Existence of Radius of Convergence of Complex Power Series/Absolute Convergence

Theorem

Let $\xi \in \C$.

Let $\ds \map S z = \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n $ be a complex power series about $\xi$.

Let $R$ be the radius of convergence of $\map S z$.

Let $\map {B_R} \xi$ denote the open $R$-ball of $\xi$.

Let $z \in \map {B_R} \xi$.

If $R = +\infty$, we define $\map {B_R} \xi = \C$.

Let $z \in \map {B_R} \xi$.

By definition of the open $R$-ball of $\xi$:

$\cmod {z - \xi} < R$

where $\cmod z$ denotes the complex modulus of $z$.

By definition of radius of convergence, it follows that $\map S z$ converges.

Suppose $R$ is finite.

Let $\epsilon = R - \cmod {z - \xi} > 0$.

Now, let $w \in \map {B_R} \xi$ be a complex number such that $R - \cmod {w - \xi} = \dfrac \epsilon 2$.

Such a $w$ exists because, if at a loss, we can always let $w = \xi + R - \dfrac \epsilon 2$.

If $R = +\infty$, then let $w \in C$ be any complex number such that $\cmod {z - \xi} < \cmod {w - \xi}$.

Then:

$\ds \limsup_{n \mathop \to \infty} \cmod {a_n \paren {z - \xi}^n}^{1/n}$

$<$

$\ds \limsup_{n \mathop \to \infty} \cmod {a_n \paren {w - \xi}^n}^{1/n}$

as $\cmod {z - \xi} < \cmod {w - \xi}$, we have $\cmod {z - \xi}^n < \cmod {w - \xi}^n$

$\ds $

$\le$

$\ds 1$

From the $n$th Root Test, it follows that $\map S z$ converges absolutely.

$\blacksquare$